Question 1090189
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<pre>
We have arithmetic progression with the first term {{{a[1]}}} = 4 and the common difference d = 3.

The number of terms, n, is unknown.


The formula for the sum of the first n terms of any AP is

{{{S[n]}}} = {{{(a[1] + ((n-1)*d)/2)*n}}}.


Thus we have an equation


{{{(4 + ((n-1)*3)/2)*n}}} = 531  ====>


(8 + 3*(n-1))*n = 1062  ====>

3n^2 + 5n - 1062 = 0  ====>


{{{n[1,2]}}} = {{{(-5 +- sqrt(5^2 + 4*3*1062))/(2*3)}}} = {{{(-5 +- 113)/6}}}.


Only positive root makes sense:  n = {{{(-5 + 113)/6}}} = 18.


<U>Answer</U>.  n = 18,  or  18 terms.
</pre>

Solved.



There is a bunch of lessons on arithmetic progressions in this site:

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Arithmetic-progressions.lesson>Arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-arithmetic-progressions.lesson>The proofs of the formulas for arithmetic progressions</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-arithmetic-progressions.lesson>Problems on arithmetic progressions</A>  

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-arithmetic-progressions.lesson>Word problems on arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-arithmetic-progressions.lesson>Mathematical induction and arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/One-characteristic-property-of-arithmetic-progressions.lesson>One characteristic property of arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problems-on-arithmetic-progressions.lesson>Solved problems on arithmetic progressions</A> 



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic <U>"Arithmetic progressions"</U>.